Math Delights

Simple Version

Our goal is to be able to determine a card chosen by a student using information about the location of the card within one of three columns.

Resources

• Math Circle Lesson Plan (Microsoft Word format)
• Poster (PDF)

Figure 1: Organizing cards aids in understanding

Execution

1. Choose 27 cards from a standard deck. Choosing specific cards and ordering them will help students understand
2. Deal the cards out in three columns of nine
3. Have the student choose one card, not revealing their choice
4. Have the student reveal the column the card is located in
5. Pick up the columns, preserving order within columns
6. Deal cards back out, from left to right
7. Repeat steps 3-6
8. Ask the student to reveal the column the card is located in
9. Reveal the card to the student

Figure 2: The choice of columns will reveal the card

This is easy to see using our ordered cards. The first time a student chooses a column, we know the suit of their card. The second time, we know the cards equivalence modulo 3. When we deal the cards out a third time, there are only three options for the student's card. When they tell you the column this time, we know their card. As stated before, this gets trickier when cards are unordered -- it's hard to recall which cards where in each column initially, and no properties are revealed by the card's location within a column.

Simple Ordering

The example below can be done using arbitrary cards; we can also replace first with second or third, with analogous results.

1. Perform the trick as usual
2. Pick up the chosen column first each time
3. Maintain order: deal out the first column first
4. Pick up cards after the column is pointed out the third time
5. The top card is the student's card

Figure 3: Picking up the students column first each time.
The student's card is moved up by one-third on each pickup.

Thank you Andrew Hermann for this card trick.